Moltiplicazione e divisione:

Siano $\displaystyle z_1=\rho_1(\cos \vartheta_1 + i \sin \vartheta_1)$ e $\displaystyle z=\rho_2(\cos \vartheta_2 + i \sin \vartheta_2)$

$\displaystyle z_1\cdot z_2=\rho_1\cdot \rho_2[\cos (\vartheta_1+\vartheta_2)+i\sin (\vartheta_1+\vartheta_2)]$

$\displaystyle \frac{z_1}{z_2} =\frac{\rho_1}{\rho_2}[\cos (\vartheta_1-\vartheta_2)+i\sin (\vartheta_1-\vartheta_2)]$

Dati $\displaystyle z_1=4\left[\cos {\left(\frac{\pi}{2}\right)}+i\sin{\left(\frac{\pi}{2}\right)}\right]$ e $\displaystyle z_2=2\left[\cos {\left(\frac{\pi}{3}\right)}+i\sin{\left(\frac{\pi}{3}\right)}\right]$

$\displaystyle z_1 \cdot z_2=8\left[\cos {\left(\frac{\pi}{2}+\frac{\pi}{3}\right)}+i\sin{\left(\frac{\pi}{2}+\frac{\pi}{3}\right)}\right]=8\left[\cos {\left(\frac{5}{6}\pi\right)}+i\sin{\left(\frac{5}{6}\pi\right)}\right]$

$\displaystyle \frac{z_1 }{z_2} =2\left[\cos {\left(\frac{\pi}{2}-\frac{\pi}{3}\right)}+i\sin{\left(\frac{\pi}{2}-\frac{\pi}{3}\right)}\right]=2\left[\cos {\left(\frac{\pi}{6}\right)}+i\sin{\left(\frac{\pi}{6}\right)}\right]$

Potenza:

Sia $\displaystyle z=\rho(\cos \vartheta + i \sin \vartheta)$

$\displaystyle z^n=\left[\rho(\cos \vartheta + i \sin \vartheta)\right]^n=\rho^n\left(\cos n\vartheta + i \sin n\vartheta\right)$ (Teorema di De Moivre)

Dato $\displaystyle z=3\left[\cos {\left(\frac{\pi}{2}\right)}+i\sin{\left(\frac{\pi}{2}\right)}\right]$

$\displaystyle z^2 =3^2\left[\cos {\left(\frac{2\pi}{2}\right)}+i\sin{\left(\frac{2\pi}{2}\right)}\right]=9\cos {\pi}+i\sin{\pi}=-9$

$\displaystyle z^5 =3^5\left[\cos {\left(\frac{5}{2}\pi\right)}+i\sin{\left(\frac{5}{2}\pi\right)}\right]=243\left[\cos {\left(\frac{\pi}{2}\right)}+i\sin{\left(\frac{\pi}{2}\right)}\right]=243i$